The generalized cone complex \(\Sigma (\mathcal {X})\)  and the generalized extended cone complex \(\overline{\Sigma }(\mathcal {X})\)  (in the terminology of [1]}) of the toroidal embedding \(\mathcal {U}\subset \mathcal {X}\)  are given by
\(\Sigma (\mathcal {X}):=\operatorname{colim}_{W_i\in \operatorname{Str}(\mathcal {X})} \sigma (W_i)\subseteq \overline{\Sigma }(\mathcal {X}):=\operatorname{colim}_{W_i\in \operatorname{Str}(\mathcal {X})} \overline{\sigma }(W_i).\) 
